484 



RICE 



ART. K 



have the same value. So it appears that if a, /3', 7' are the 

 direction cosines of any one of the three principal axes then 



aia + ae/S' + a^y' 

 aea + a2/3' + 0*7' 

 asa' + ttifi' + 037' 



pa', 



pt', 



where p is a multiplier still undetermined, but the same in all 

 three equations. These, combined with the equation 

 Q,'2 _j_ ^'2 _|_ y'2 = X, are sufficient to determine, first the value of 

 p, and then the values of a', ^', y' in terms of the six strain-func- 

 tions, tti, 02, ... a 6. The analysis is exactly similar to that 

 which we employed earlier when explaining the conditions for 

 the existence of a homogeneous strain in a solid in contact with a 

 liquid. We write the preceding equations in the form 



(ai — p)a + ae/S' + 057' 

 aea' + (a2 - p)l3' + 047' 

 a^a' + a^jQ' + (as - p)7' 



[429a] 



(The reader will easily satisfy himself that these are the equa- 

 tions [429] with p substituted for rl) Now, for reasons which we 

 have already discussed in the place just referred to, these three 

 equations are not consistent with one another unless the follow- 

 ing determinantal equation is true: 



as 



O3 — P 



= 0, 



and this is actually equation [430], with p substituted for r^. 

 It is of course a cubic equation in p and can be written, on 

 expanding the determinant, as 



Ep^ -\- Fp - G = 0, 



where 



E 

 F 



ai + a2 + as, 

 a2a3 + azai + aia2 



ai^ — as^ 



a6^ 



